We consider in this paper the solvability of linear integral equations on the real line, in
operator form (λ − K)φ = ψ, where λ ∈ C and K is an integral operator. We impose
conditions on the kernel, k, of K which ensure that K is bounded as an operator on
X := BC(R). Let Xa denote the weighted space Xa := {χ ∈ X: χ(s) = O(|s|−a ) as
|s| →∞}. Our first result is that if, additionally, |k(s, t)| κ(s − t), with κ ∈ L1(R) and
κ(s) = O(|s|−b) as |s|→∞, for some b >1, then the spectrum of K is the same on Xa as
on X, for 0
1. As an example where kernels of this
latter form occur we discuss a boundary integral equation formulation of an impedance
*boundary value problem for the Helmholtz equation in a half-plane.
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